Expressing Natural Numbers as Sum of Primes

[metrictensor]

Is it possible to express all natural numbers greater than 2 as the sum of N unique prime numbers? For example, 6 = 2 + 3 and 18 = 13 + 5.

 

[matt grime]

6=2+3 does it?

What about the smallest natural greater than 2 that isn't a prime? isn't that a counter example too?

 

[metrictensor]

6=2+3 does it?

What about the smallest natural greater than 2 that isn't a prime? isn't that a counter example too?

I forgot to mention that 1 and 2 can be used in the sum. I made a mistake. Not 6 = 2 + 3, 5 = 2 + 3. Once you define 1 and 2 you get 3. From 3+1 you get 4 and so on. 5=2+3...

 

[HallsofIvy]

So you are NOT talking about a "sum of primes" because 1 is not prime.

 

[matt grime]

Now I'm a little perplexed. 2 is a prime. Anyway, the result follows quite easily from Russell's postulate:

For every n>1 there is a prime number p such that n<p<2n.

So, given m an integer, if it is even pick an prime in the range m/2 to m, and if it is odd pick a prime in the range (m+1)/2 to m+1, call this prime p(1).

Then m-p(1) < m/2, and we proceed by indcuction - the next prime we pick must be distinct from p(1) since it must be less than m/2, and p(1) is greater than m/2.

We just need to show how to write the sum for "small m" to get the full proof, ie m=1,2,3 which yo'uve done.

 

[robert Ihnot]

Matt grime: the result follows quite easily from Russell's postulate.

Yeah? As re-expressed by Euler, an equivalent form of this conjecture (called the "strong" or "binary" Goldbach conjecture) asserts that all positive even integers can be expressed as the sum of two primes.http://mathworld.wolfram.com/GoldbachConjecture.html

So you are saying the Goldbach Conjecture has been proven?

 

[Icebreaker]

Schnirelman (1939) proved that every even number can be written as the sum of not more than 300 000 primes (Dunham 1990)

What if we added up the first 300 002 primes?

 

[Zurtex]

What if we added up the first 300 002 primes?

What is that supposed to mean ?

 

[Icebreaker]

Nevermind.

 

[matt grime]

Matt grime: the result follows quite easily from Russell's postulate.

Yeah? As re-expressed by Euler, an equivalent form of this conjecture (called the "strong" or "binary" Goldbach conjecture) asserts that all positive even integers can be expressed as the sum of two primes.http://mathworld.wolfram.com/GoldbachConjecture.html

So you are saying the Goldbach Conjecture has been proven?

Of course not, but the OP didn't ask for the sum of TWO primes, he asked for the sum to be expressed as *some* sum of N distinct primes, allowing for 1 to be included.

The possibly dodgy assumption I made was that it wasn't supposed to be a fixed N, which seems reasonable, given the other hidden assumptions in the post, after all it would be required that N=1 or 2 otherwise, which we know to be false and bloody hard respectively.

 

[robert Ihnot]

Matt Grime: but the OP didn't ask for the sum of TWO primes.

Oh, that is correct! I just followed his examples, you know: 6=2+3 and 18 =13+5. But do you mean, Bertrand/s postulate?
http://mathforum.org/library/drmath/view/51505.html

 

[matt grime]

Well, whatever it should be called, it states given any natural n greater than 2 there is a prime p satisfying n<p<2n, or similar.

 

[Zurtex]

Well, whatever it should be called, it states given any natural n greater than 2 there is a prime p satisfying n<p<2n, or similar.

It has been proved there exists a prime for any natural number n > 2 there exists a prime (and I'm really dragging this one from memory) p such that:

$$n - n^{\frac{23}{42}} < p < n$$

Which is quite a bit stronger . Although I'm not sure how useful.

 

[CRGreathouse]

Here's the formula on the Wolfram site

It has been proved there exists a prime for any natural number n > 2 there exists a prime (and I'm really dragging this one from memory) p such that:

$$n - n^{\frac{23}{42}} < p < n$$

Which is quite a bit stronger . Although I'm not sure how useful.

http://functions.wolfram.com/NumberTheoryFunctions/Prime/29/0003/